So let's get on with copying everything from the old hard drive to the new one. You can't just drag-and-drop copy the files, as that will miss your boot sector, important parts of Windows, and any hidden partitions. If your computer came with Windows installed, there's a good chance that you've got a hidden partition that you'll need on the new drive should you have to reinstall the OS.
There are two approaches here: Imaging and cloning. Imaging backs up everything on your hard drive to a single, very large file on a separate piece of media--usually an external hard drive. Cloning directly copies the contents of one drive onto another. I'm going to discuss cloning here because, for this particular job, it's probably the better choice.
Cloning requires that your PC have access to both hard drives simultaneously. If you have a desktop, you can install the new one as a second internal drive (which will become the first one after you remove the older drive). If you have a laptop, or if you just don't want to fiddle with motherboard cables anymore than you have to, you can use a device like the BYTECC USB 2.0 to IDE/SATA Adapter to turn your new internal hard drive into a temporary external one.
For cloning software, consider EASEUS Disk Copy. This free program downloads as a .iso image file, which you can use to create a bootable CD. (If double-clicking the .iso file doesn't launch a disc-burning program, download and install the free ISO Recorder.) When you boot from the CD, it will take you into a simple wizard-driven cloning program.
Disk Copy misses one important feature: You can't resize the partitions, an odd omission since the whole point of upgrading your hard drive is to get a bigger one. Luckily, if you have Windows 7 or Vista, that shouldn't be too much of a problem.
If you're using XP, I recommend the EASEUS Partition Master Home Edition (yes, same company). And not only for resizing the partitions. In addition to being an excellent and free partition manager, it can also clone drives. Just select your source drive (the old one) and click the Copy button on the top toolbar for a simple and friendly wizard.
Oddly, this wizard also doesn't let you enlarge partitions. But then, once the job is done, it's easy enough to enlarge your partitions in the main EASEUS program.
Unfortunately, EASEUS Partition Master Home Edition won't work in 64-bit environments. You don't need it for Vista or Windows 7, since you can use EASEUS Disk Copy and Windows' own partition-resizing tools.